Fact 2.3 [1]} 
Suppose that \(f \in \Gamma _{0} (\mathcal {H}) \) . Then \(f\)  is Legendre if and only if \(f^{*}\)  is. In this case, the gradient mapping \(\nabla f: \operatorname{int}\operatorname{dom}f \rightarrow \operatorname{int}\operatorname{dom}f^{*}\) 
is bijective, with inverse \((\nabla f)^{-1} =\nabla f^{*} : \operatorname{int}\operatorname{dom}f^{*} \rightarrow \operatorname{int}\operatorname{dom}f\) . Moreover, the gradient mappings \(\nabla f\) , \(\nabla f^{*}\)  are both norm-to-weak continuous and locally bounded on their respective domains.

